Why is the solar system stable?

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: lawrence@krubner.com, or follow me on Twitter.

I am fascinated with the question of smart people thinking dumb things. Or rather, things that now strike me as stupid, partly because I grew up knowing the answer.

One of the smartest people who ever lived was Isaac Newton. And for a long time, he was convinced that the sun had a repulsive force that was pushing the planets away. Robert Hooke had to expend considerable effort to convince Newton that the sun had an attractive force. And then, once Newton converted to that view, Newton’s thoughts went through a 2 step evolution:

1.) he published De Moto in 1685, but he ends the book in deep despair, saying that the sun and the planets all attract each other, but that no one will ever be able to figure out a convenient method of calculating those forces of attraction. Indeed, he says that to do so is beyond the ability of the human race.

2.) he published Principia Mathematica in 1687, in which says that all objects attract all other objects, proportionally to their mass and inversely proportional to the square of the distance between them, and he called this attractive force “gravity”.

I grew up with the concept of gravity so to me it seems obvious. But that raises the issue, why did Newton go most of his life believing the wrong thing? Was he stupid, or what?

Clearly, he wasn’t stupid. He was a lot smarter than me. So why did he have so much trouble getting the right answer?

I’ve tried to put myself in his shoes, and I realize, based on what he wrote, that one of his big concerns was the stability of the solar system. Why is it so damn stable? And in fact, the more I think about this, the weirder it seems. If you believe that the Sun has an attractive force, then you are saying that you believe that the Earth just happens to have exactly the right mass and momentum to be in the orbit it is right now. If it was slightly faster then it would drift away from the Sun and eventually leave the Solar System. If it was slightly slower then it would have eventually fallen into the Sun. That the Earth just happens to be in this exact orbit is a very large coincidence.

In the modern world, we are helped by the belief that the early Solar System had millions of objects of all shapes and sizes, going at different speeds and in different orbits, and that all the objects that were going to fall into the Sun already did fall into the Sun, and all of the objects that were going to leave the Solar System have already left the Solar System, and the Earth just happens to be one of the 8 objects that survives, out of those initial millions, that had a “just-so” orbit that leaves it stable. Newton lacked this belief about the early Solar System, so for him, the stability of the Solar System must have struck him as a coincidence too large for a reasonable person to accept. He assumed God must still be managing the Solar System.

I was thinking of this and then I ran into this article about the stability of the Solar System:

The most straightforward way to solve the problem of the stability of the solar system is to follow the planetary orbits for a few billion years on a computer. All of the planetary masses and their present orbits are known very accurately and the forces from other bodies—passing stars, the Galactic tidal field, comets, asteroids, planetary satellites, etc.—are either easy to incorporate or extremely small. There are two main challenges. The first is to devise numerical methods that can follow the motions of the planets with sufficient accuracy over a few billion orbits; this was solved by the development in the 1990s of symplectic integration algorithms, which preserve the geometrical structure of dynamical flows in multidimensional phase space and thereby provide much better long-term performance than general-purpose integrators.The second challenge was the overall processing time needed to follow planetary orbits for billions of years; this was solved by the exponential growth in speed of computing hardware that has persisted for the last five decades. At the present time, following planetary systems over billion-year intervals is difficult mostly because it is a serial problem—you have to follow the orbits from 2011 to 2020 before you can follow them from 2021 to 2030—whereas most of the computational speed gains of the last few years have been achieved by parallelization, the distributing of a computing problem among hundreds or thousands of processors that work simultaneously.

So what are the results? Most of the calculations agree that eight billion years from now, just before the Sun swallows the inner planets and incinerates the outer ones, all of the planets will still be in orbits very similar to their present ones. In this limited sense, the solar system is stable. However, a closer look at the orbit histories reveals that the story is more nuanced. After a few tens of millions of years, calculations using slightly different parameters (e.g., different planetary masses or initial positions within the small ranges allowed by current observations) or different numerical algorithms begin to diverge at an alarming rate. More precisely, the growth of small differences changes from linear to exponential: at early times, the differences in position at successive time intervals grow as 1 mm, 2 mm, 3 mm, etc., while at later times they grow as 1 mm, 2 mm, 4 mm, 8mm, 16 mm, etc. This behavior is the signature of mathematical chaos, and implies that for practical purposes the positions of the planets are unpredictable further than about a hundred million years in the future because of their extreme sensitivity to initial conditions. As an example, shifting your pencil from one side of your desk to the other today could change the gravitational forces on Jupiter enough to shift its position from one side of the Sun to the other a billion years from now. The unpredictability of the solar system over very long times is of course ironic since this was the prototypical system that inspired Laplacian determinism.

Fortunately, most of this unpredictability is in the orbital phases of the planets, not the shapes and sizes of their orbits, so the chaotic nature of the solar system does not normally lead to collisions between planets. However, the presence of chaos implies that we can only study the long-term fate of the solar system in a statistical sense, by launching in our computers an armada of solar systems with slightly different parameters at the present time—typically, each planet is shifted by a random amount of about a millimeter—and following their evolution. When this is done, it turns out that in about 1 percent of these systems, Mercury’s orbit becomes sufficiently eccentric so that it collides with Venus before the death of the Sun. Thus, the answer to the question of the stability of the solar system—more precisely, will all the planets survive until the death of the Sun—is neither “yes” nor “no” but “yes, with 99 percent probability.”

There remain two intriguing facts that lead to a plausible speculation. First, the future time required for the loss of Mercury is rather similar, within a factor of five or so, to the past time at which the solar system was born. Second, the solar system is nearly “full,” in the sense that there are few places where we could insert an additional planet without causing immediate instability. Both of these facts are explained naturally if the solar system began with more planets, in a configuration that was unstable on timescales much smaller than its current age. As time passed, the system lost more and more planets and thereby gradually self-organized into a more and more stable state. In this process, the time required to lose the next planet would quite naturally be a few times the current age. There would be few fossil traces of these lost siblings of the Earth.

Post external references

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    http://www.ias.edu/about/publications/ias-letter/articles/2011-summer/solar-system-tremaine
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